In the regression model $\mathbf{Y} = \eta + \mathbf{BX} +
\mathbf{Z}$ with $\mathbf{Z}$ unobserved, $\mathscr{E}\mathbf{Z} = \mathbf{0}$
and $\mathscr{E}\mathbf{ZZ}' = \mathbf{\Sigma}_{ZZ}$, the least squares
estimator of $\mathbf{B}$ is $\hat{\mathbf{B}} =
\mathbf{S}_{YX}\mathbf{S}_{XX}^{-1}$. If the rank of $\mathbf{B}$ is known to
be $k$ less than the dimensions of $\mathbf{Y}$ and $\mathbf{X}$, the reduced
rank regression estimator of $\mathbf{B}$, say $\mathbf{B}_k$, depends on the
first $k$ canonical variates of $\mathbf{Y}$ and $\mathbf{X}$. The asymptotic
distribution of $\hat{\mathbf{B}}_k$ is obtained and compared with the
asymptotic distribution of $\hat{\mathbf{B}}$. The advantage of
$\hat{\mathbf{B}}_k$ is characterized.